Reliability-Based Load and Resistance Factor Design (LRFD) Guidelines for Stiffened Panels and Grillages of Ship Structures

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1 Reliability-Based Load and Resistance actor Design (LRD) Guidelines or Stiened Panels and Grillages o Ship Structures Ibrahim A. Assakka 1, Bilal M. Ayyub 2, Paul E. Hess, III, 3 and Khaled Atua 4 ABSTRACT Stiened panels and grillages are very important components in ship and oshore structures, and thereore they should be designed or a set o ailure modes that govern their strength. They orm the backbone o most ship s structure, and they are by ar the most commonly used element in a ship. They can be ound in bottom structures, decks, side shell, and superstructures. To evaluate the strength o a stiened panels or grillages, it is necessary to review various strength predicting models and to study their, biases, applicability, and limitations or dierent loading conditions acting on the element. In this paper, strength limit states or various ailure modes o ship panels are presented. or each limit state, commonly used strength models were collected rom many sources or evaluating their limitations and applicability and to study their biases and uncertainties. Wherever possible, the dierent types o biases resulting rom these models were computed. The bias and uncertainty analyses or these strength models are needed or the development o load and resistance actor design (LRD) guidelines or stiened panels and grillages o ship structures. The uncertainty and biases o these models were assessed and evaluated by comparing their predictions with ones that are more accurate or real values. The objective o this paper is to develop load and resistance actor design (LRD) or stiened panels and grillages o ship structures. Monte Carlo simulation was used to assess the 1 Director o Reliability Research, Center or Technology and Systems Management, Department o Civil and Environmental Engineering, University o Maryland, College Park Contact author, Proessor & Director, Center or Technology & Systems Management, Department o Civil & Environmental Engineering, University o Maryland, College Park, MD 20742, (Tel), ayyub@umail.umd.edu 3 Structures and Composites Department, Naval Surace Warare Center - Carderock Division, Code 65, West Bethesda, MD Senior Reliability Engineer, KLA Tencor, CA, and Assistant Proessor, Department o Naval Architecture, Alexandria University, Egypt. 1

2 biases and uncertainties or these models. Recommendations or the use o the models and their biases in LRD development are provided. The irst-order reliability method (ORM) was utilized to develop the partial saety actors (PS s) or selected limit states. 1. INTRODUCTION The main type o raming system ound in ships nowadays is a longitudinal one, which has stieners running in two orthogonal directions (igure 1). Deck and bottom structures panels are reinorced mainly in the longitudinal direction with widely spaced heavier transverse stieners. The main purpose o the transverse stieners is to provide resistance to the loads induced on bottom and side shell by water pressure. The types o stieners used in the longitudinal direction are the T-beams, angles, bulbs, and lat bars, while the transverse stieners are typically T-beam sections. This type o structural coniguration is commonly called gross stiened panel or grillage (Vroman, 1995). Besides their use in ship structures, these grillages are also widely used in land-based structures such as box and plate girders. A typical longitudinal stiened sub-panel, as shown in igure 1, is bounded on each end by a transverse structure, which has signiicantly greater stiness in the plane o the lateral load. The sides o the panel are deined by the presence o a large structural member that has greater stiness in bending and much greater stiness in axial loading. In ship structures, there are three types o loading that can eect the strength o a platestiener panel; negative bending moment, positive bending moment, and in-plane compression or tension. Negative bending loads are the lateral loads due to lateral pressure. They cause the plate to be in tension and the stiener lange to be in compression. Positive bending loads are those loads that put the plating in compression and the stiener lange in tension. The third type o loading is uniorm in-plane compression. This type o loading arises rom hull girder bending, and will be considered to be positive when the panel is in compression. The three types o loading can act individually or in combination with one another. To evaluate the strength o a stiened panels and grillages element, it is necessary to review various strength prediction models and to study their applicability and limitations or dierent loading conditions acting on the element. The uncertainties that are associated with a numerical analysis are generally a result o experimental approximation or numerical inaccuracies that can be reduced by some procedures. However, the uncertainties that are 2

3 associated with a strength design model are dierent and cannot be eliminated because they result rom not accounting or some variables, which can have strong inluence on the strength. or this reason, the uncertainty and the bias o a design equation should be assessed and evaluated by comparing its predictions with more accurate ones. Wherever possible, the dierent types o biases resulting rom these models were computed. In doing so, these prediction models were classiied as ollows (Atua and Ayyub, 1996): (1) prediction models that can be used by the LRD guidelines, (2) advanced prediction models that can be used or various analytical purposes, (3) some experimental results rom model testing, and (4) some real measurements based on ield data during the service lie o a ship. urthermore, the relationships and uncertainty analyses or these models are required. The relationships can be deined in terms o biases (bias actors). In this paper, only selected strength models that are deemed suitable or LRD design ormat are highlighted and presented. 2. DESIGN LOADS AND LOAD COMBINATIONS Primary structural loads on a ship are due to its own weight, cargo, buoyancy, and operation in a random environment, i.e., the sea. The loads acting on the ship s hull girder can be categorized into three main types that are used in this paper: (a) stillwater loads, (b) wave loads, and (c) dynamic loads. The load eect o concern herein is bending moment exerted on the ship hull girder. Hydrostatic lateral pressure on stiened plates (panels) is due to several sources that include: stillwater, wave and dynamic eects, green seas, and liquids in tanks. Only the irst two types are considered in this paper. Mansour et al (1996) assumed coeicients o variation ( COV s) o 0.2 and 0.1 or stillwater and wave-induced pressures. In this paper, the COV or stillwater pressure is assumed to be 0.15, the COV or wave-induced pressure is 0.15, the COV or dynamic-induced pressure is 0.25, and the COV or the combined wave and dynamic-induced pressure is These values were selected based on judgment. Stillwater loads can be predicted and evaluated with a proper consideration o variability in weight distribution along the ship length, variability in its cargo loading conditions, and buoyancy. Both wave loads and dynamic loads are related and aected by many actors such as ship characteristics, speed, heading o ship at sea, and sea state (waves heights). Waves height is a random variable that requires statistical and extreme analyses o ship response data collected over a period o time in order to estimate maximum wave-induced and dynamic bending 3

4 moments that the ship might encounter during its lie. The statistical representation o sea waves allows the use o statistical models to predict the maximum wave loads in ship s lie. Procedures or computing design wave loads or a ship s hull girder based on spectral analysis can be ound in numerous reerences pertaining to ship structures such as Hughes (1988) and Ayyub et al. (2002b). 2.1 Design Loads The design loads that are o concern in this study or developing reliability-based design or stiened panels and grillages o ship structures are those loads resulting rom ship hull girder bending and their combinations. As indicated earlier, the loads acting on the ship s hull girder can be categorized into three main types: stillwater loads, wave loads, and dynamic loads. Each o these types o loads is described in detail in Assakka et al. (2002). These are the same types o loads used or the development o LRD guidelines or unstiened panels in Assakka et al. (2002). 2.2 Load Combinations and Ratios Reliability-based structural design o stiened panels and grillages as presented in this paper is based on two load combinations that are associated with correlation actors as presented in the subsequent sections (Mansour et al. 1984) Stillwater and Vertical Wave-induced Bending Moments The load eect (stress) on stiened panel element due to combinations o stillwater and vertical wave-induced bending moments is given by = + k (1) c WD WD where = stress due to stillwater bending moment, WD = stress due to wave-induced bending moment, c = un-actored combined stress, k W = correlation actor or wave-induced bending moment and can be set equal to one (Mansour et al. 1984) Stillwater, Vertical Wave-induced, and Dynamic Bending Moments The load eect on stiened panel element due to combinations o stillwater, vertical wave-induced and dynamic bending moments is given by 4

5 = + k ( + k ) D (2) c W where = stress due to stillwater bending moment, W = stress due to waves bending moment, D = stress due to dynamic bending moment, c = un-actored combined load, and k D = correlation actor between wave-induced and dynamic bending moments. The correlation actor k D is given by the ollowing two cases o hogging and sagging conditions (Mansour et al and Atua 1998): a. Hogging Conditions: W D Exp ( 158LBP LBP ) LBP k = D 3 (3) b. Sagging Condition: Exp ( 158LBP LBP ) LBP k = D 3 (4) where LBP = length between perpendiculars or a ship in t. Values o k D or LBP ranging rom 300 to 1000 t can be obtained either rom Table 1 or rom the graphical chart provided in igure LIMIT STATES AND DESIGN STRENGTH The stiened panel o ship structure or all stations should meet one o the ollowing conditions; the selection o the appropriate equation depends on the availability o inormation as required by these two equations: φ γ + γ k (5) u WD WD WD where u φ φ γ + k ( γ + γ k ) D (6) u = ultimate strength (stress) or stiened panel, W W = strength reduction actors or ultimate strength capacity o a stiened panels and grillages, γ = load actor or the stress due to stillwater bending moment W D D 5

6 = stress due to stillwater bending moment k WD = combined wave-induced and dynamic bending moment actor γ WD = load actor or the stress due combined wave-induced and dynamic bending moment WD = stress due to combined wave-induced and dynamic bending moments k W = load combination actor, can be taken as 1.0 γ W = load actor or the stress due waves bending moment W = stress due to waves bending moment k D = load combination actor, can be taken as 0.7 γ D = load actor or the stress due to dynamic bending moment D = stress due to dynamic bending moment The nominal (i.e., design) values o the strength and load components should satisy these ormats in order to achieve speciied target reliability levels. The nominal strength or stiened panels can be determined as described in subsequent sections. 3.1 Design Strength or Stiened Panels and Grillages Stieners Stieners are very important structural components that are used to strengthen plates and to increase their load carrying capacity. In ship structures, most o grillage ailures are due to the collapse o one or more o the longitudinal and transverse stieners. Thus, the irst and most basic principle with regard to stieners is that they should be designed at least as strong as the plating. Also, they should be suiciently rigid and stable so that neither local stiener buckling nor overall buckling occurs beore local plate buckling. A plate stiener can be subjected to a variety o primary and secondary loads and load combinations that cause the stiened plate to ail in one o the ollowing types o buckling: (1) column buckling, (2) beam-column buckling, and (3) lexural-torsional buckling. Numerous strength models or stieners are available 6

7 according to the type o stiener buckling involved, and can be ound in API (1993), Assakka (1998), and Atua (1998) Longitudinal Strength o Stiened Panels In this section, a summary o selected strength models that are deemed suitable or LRD design ormats is presented. These strength models are or longitudinally stiened panels subjected to various types o loading. They are presented herein in a concise manner, and they were evaluated in terms o their applicability, limitations, and biases with regard to ship structures. A complete review o the models used by dierent classiication agencies such as the AISC (1994), ASSHTO (1994), and the API (1993) is provided in Atua (1998) and Assakka (1998) Herzog s Model Based on reevaluation o 215 tests by various researchers and on empirical ormulation, Herzog (1987) developed a simple model (ormula) or the ultimate strength o stiened panels that are subjected to uniaxial compression without lateral loads. The ultimate strength u o a longitudinally stiened plate is given by the ollowing empirical ormula (Herzog 1987): u m = m y y ka rπ ka rπ E y E y b t or or b t b t 45 > 45 (7) where y = ys A A s s + + A yp p A p, mean yield strength or the entire plate-stiener cross section yp ys E A p A s A = yield strength o plating = yield strength o stiener = modulus o elasticity o stiened panel = bt, cross sectional area o plating = t w + t w d w, cross sectional area o stiener = A s + A p, cross sectional area o plate-stiener 7

8 t w t w d w a b t I = stiener lange thickness = stiener lange width or breadth = stiener web thickness = stiener web depth = length or span o longitudinally stiened panel = distance between longitudinal stieners = plate thickness = moment o inertia o the entire cross section r = A I, radius o gyration o entire cross section m = corrective actor accounts or initial deormation and residual stresses k = buckling coeicient depends on the panel end constraints Values or m and k or use in Eq. 6-7 can be obtained rom Tables 2 and 3, respectively. The 215 tests evaluated by Herzog belong to three distinct groups. Group I (75 tests) consisted o small values or imperection and residual stress, Group II (64 tests) had average values or imperection and residual stress, while the third group (Group III, 76 tests) consisted o higher values or imperection and residual stress. The statistical uncertainty (COV) associated with Herzog model o Eq. 7 is The mean value µ, standard deviation σ, and COV o the measurement to prediction are given in Table Hughes s Model According to Hughes (1988), there are three types o loading that must be considered or determining the ultimate strength o longitudinally stiened panels. These types o loading are: (1) Lateral load causing negative bending moment o the plate-stiener combination (the panel), (2) Lateral load causing positive bending moment o the panel, and (3) In-plane compression resulting rom hull girder bending. The sign convention to be used throughout this section is that o Hughes (1988). Bending moment in the panel is considered positive when it causes compression in the plating and tension in the stiener lange, and in-plane loads are positive when in compression (igure 3). The delection, w 0, due to the lateral load (i.e., lateral pressure) M 0 and initial eccentricity, δ 0, are considered positive when they are toward the stiener as shown in igure 3. In beam-column theory, the expressions or the moment M 0 and the 8

9 corresponding delection w 0 are based upon an ideal column, which is assumed to be simply supported. Disregarding plate ailure in tension, there can be three distinct modes o collapse (see igure 3) according to Hughes (1988): (1) Compression ailure o the stiener (Mode I Collapse), (2) Compression ailure o the plating (Mode II Collapse), and (3) Combined ailure o stiener and plating (Mode III Collapse). The ultimate axial strength (stress) u or a longitudinally stiened panel under a combination o in-plane compression and lateral loads (including initial eccentricities) can be, thereore, deined as the minimum o the collapse (ultimate) values o applied axial stress computed rom the expressions or the three types (modes) o ailure. Mathematically, it can be given as = min(,, ) (8) u a, ui a, uii a, uiii where a,ui, a,uii, and a,uiii correspond to the ultimate collapse value o the applied axial stress or Mode I, Mode II, and Mode III, respectively. The mathematical expressions or the collapse stress or each mode o ailures are provided in Hughes (1988) Adamchak s Model Adamchak developed this model in 1979 to estimate the ultimate strength o conventional surace ship hulls or hull components under longitudinal bending or axial compression. The model itsel is very complex or hand calculation and thereore it is not recommended or use in a design code without some computational tools or a computer program. To overcome the computational task or this model, Adamchak developed a computer program (ULTSTR) based on this model to estimate the ductile collapse strength o conventional surace ship hulls under longitudinal bending. The recent version o the ultimate strength (ULTSTR) program is intended or preliminary design and based on a variety o empirically based strength o material solutions or the most probable ductile ailure modes or stiened and unstiened plate structures. The probable ductile ailure modes include section yielding or rupture, inter-rame Euler beamcolumn buckling, and inter-rame stiener tripping (lateral-torsional buckling). The program also accounts or the eects o materials having dierent yield strength in plating and stieners, or initial out-o-plane distortion due to abrication, and or lateral pressure loading. The basic theory behind this model (or ULTSTR) originated preliminary in a joint project on ship structural design concepts involving representatives o the Massachusetts Institute o 9

10 Technology (MIT), the Ship Structure Committee (SSC), and navy practices in general. Longitudinally stiened panel elements can ail either by material yielding, material rupture (tension only), or by some orm o structural stability. The instability ailure modes or this model include Euler beam-column buckling and stiener lateral torsional buckling (tripping). These modes o ailure are illustrated in igure 4. Euler beam-column buckling is actually treated in this model as having two distinct types o ailure patterns as shown in igure 5. Type I is characterized by all lateral deormation occurring in the same direction. Although this type o ailure is depended on all geometrical and material properties that deine the structural element, it is basically yield strength dependent. Type I ailure is assumed to occur only when either lateral pressure or initial distortion, or both, are present. On the other hand, Type II ailure is modulus (E) depended, as ar as initial buckling is concerned. This type o ailure can be initiated whether or not initial distortion or lateral pressure, or both, are present. Type III ailure is a stiener tripping or lateral-torsional buckling. Thereore, the ultimate axial strength (stress) or longitudinally stiened panel under various types o loading (including material abrication distortion) is the minimum value o the axial compressive stress computed rom the expressions or the three types (modes) o ailures, that is = min(,, ) (9) u ui Detailed mathematical expressions or the three modes o ailures as implemented in the program ULTSTR can be ound in Adamchak (1979) Paik and Lee s Model An empirical ormula, developed by Paik and Lee (1996), or predicting the ultimate strength o longitudinal stiened sub-panels based on 130-collapse test data or stiened plates with initial imperections is presented. The ormula expresses the ratio o ultimate strength o the sub-panels to its yield strength in terms o the plate slenderness ratio, β, and the stiener slenderness ratio, λ, as ollows: uii uiii [ λ B λ B 0. ] u = λ (10) y( panel ) 067 where y(panel) = yield strength o the whole panel and is given by 10

11 ypζys y( panel) = (11) 1+ ζ where d w tw + wt ζ = bt (12) The plate slenderness ratio, B, is given by b y B = t E (13) The stiener slenderness ratio, λ, is given by a y λ = πr E (14) in which, a = span (length) o stiener, r = radius o gyration o one stiener with ully eective plating and is given by r = I A (15) where A = sectional area o the plate and the stiener and it is given by A = bt + d t + t (16) w w w The moment o inertia o one stiener with ully eective plating (I) is given by bt t d wtw d wt t w I = + bt z + + d + + wtw z t wt z t d w (17) where z 0 = distance o neutral axis rom the base line o plate, t = thickness o plate, t w = thickness o stiener web, t = thickness o stiener lange, d w = stiener web height, b = spacing between stiener, and w = stiener lange width. The ormula was compared with experimental and numerical data (Paik and Lee 1996, and Paik 1997) and proved to predict the strength value reasonably. 2 11

12 AASHTO The ultimate strength o a stiened panel subjected to uniaxial compressive strength is given by (AASHTO 1994) where ( 0.66) λ y or λ 2.25 u = 0.88 (18) y or λ > 2.25 λ The limiting width/thickness ratios or axial compression is to satisy 2 ak y λ = (19) rπ E b E k (20) t y where b = spacing between stiener, a = length o panel, E = Young s Modulus, y = weighted yield strength, and k = plate buckling coeicient as speciied in Table Gross Panels and Grillages To perorm a reliability (saety) checking on the design o grillages, the ratio o the stiness o the transverse and longitudinal stieners should at least equal not be the load eect given by the geometrical parameters shown in the second hand term o the ollowing expression: I I y x = ( n + 1) 2 2 nπ N 5 3 b a 5 (21) where I x = moment o inertia o longitudinal plate-stiener, I y = moment o inertia o transverse plate-stiener, a = length or span o the panel between transverse webs, b = distance between longitudinal stieners, n = number o longitudinal stieners, and N = number o longitudinal sub-panels in overall (or gross) panel. A target reliability level can be selected based on the ship type and usage. Then, the corresponding saety actor can be looked up rom Table 6. 12

13 4. COMPARISON AND EVALUATION O EXISTING MODELS OR STIENED PANELS In this section, a comparison between real and predicted values o ultimate strength was perormed based on real test specimens rom various sources. Some o the strength models used in this comparison are adopted in the current design codes such as the AISC LRD (1994), AASHTO LRD (1994), and API (1993). Other models that are also used in this comparison, are those developed by dierent researchers such as Hughes (1988), Adamchak (1997), Herzog (1987), Paik and Lee (1996), and Mikami and Niwa (1996). The purpose o this comparison is to select the most appropriate model (models) or use in LRD design ormat. The level o complexity associated with the above-mentioned strength models ranges rom highly complex models to simple ones. The more complex theoretical models, such as that o Hughes (Eq. 8) and Adamchak (Eq. 9), do not necessarily lead to less uncertainty. Although they can be accurate and rigorous models, they can lead to more uncertainty because they involve a larger number o variables, some o which may be very uncertain. On the other hand, simple empirical ormulations based on real test data, such as that o Herzog (Eq. 7) and Paik and Lee (Eq. 10), can lead to airly good results. Although theoretically less rigorous, they can be o practical use because they were derived rom real world stiened plates tests. In ormulating a design model, a balance must be achieved between the model accuracy, bias, applicability, and simplicity, all o which are desired eatures. Uncertainty and bias o a strength model can be assessed by comparing its strength prediction with a model that has more accurate result, or real value. In the subsequent sections, bias assessments or uniaxial strength o longitudinally stiened panels under axial and lateral (pressure) loads are presented. 4.1 Bias Assessment or Uniaxial Strength Models without Lateral Pressure This section summarizes the results o comparisons that were perormed by Assakka and Atua (1997) on nearly 80 test specimens under uniaxial load alone. The ailure axial stress and the mode o ailure or each test were reported. Table 7 provides the mean, standard deviation, 13

14 and the COV o the bias (real / predicted) or Hughes (1988), Herzog (1987), Adamchak (1997), and Paik and Lee (1996) models. It is apparent rom the results in this table that these models have the least bias values or the predicted strength (stress). Table 8 gives the mean, standard deviation, and the COV o the bias (real / predicted) or the strength models used in the current design codes or stiened panels. Variations in the bias as a unction o column slenderness ratio or Hughes (1988), Herzog (1987), Adamchak (1997), and Paik and Lee (1996) models are shown in igure 6. igure 7 gives the variation in the bias or the current design codes. 4.2 Bias Assessment or Uniaxial Strength Models with Lateral Pressure Assakka (1998) and Atua (1998) perormed comparison analyses on 14 test specimens subjected to a combination o uniaxial stress and uniorm lateral pressure. or each test, they reported the ailure axial stress and the mode o ailure. Table 9 gives the mean, standard deviation, and the COV o the bias (real / predicted) or Hughes (1988), Herzog (1987), Adamchak (1997), and Paik and Lee (1996) models. The results in this table suggest that these models have the least bias values or the predicted ultimate strength (stress) as compared to the values predicted by the codes. Table 10 gives the mean, standard deviation, and the COV o the bias (real / predicted) or the strength models used in the current design codes or stiened panels. Variations in the bias as a unction o the ratio o applied moment to plastic moment or stiened panels with simply supported ends are shown in igure 8. igure 9 gives the variations in the bias or the clamped case. 5. LRD GUIDELINES OR STIENED PANELS AND GRILLAGES 5.1 Target Reliability Levels Selecting a target reliability level is required in order to establish reliability-based design guidelines or ship structures such as the stiened panels and grillages. The selected reliability 14

15 level determines the probability o ailure o the stiened panels and grillages. The ollowing three methods can be used to select a target reliability value: 1. Agreeing upon a reasonable value in cases o novel structures without prior history. 2. Calibrating reliability levels implied in currently used design codes. 3. Choosing a target reliability level that minimizes total expected costs over the service lie o the structure or dealing with design or which ailures result in only economic losses and consequences. The recommended range o target reliability indices or stiened panel can be set to range rom 3.0 to 4.0 (Mansour et al. 1996), while or grillage it ranges rom 2.0 to Statistical Characteristics o Basic Random Variables The statistical characteristics o random variables o strength and load models are needed or reliability-based LRD and assessment o ship structures including stiened panels. The moment methods or calculating partial saety actors (Ang and Tang 1990, Ayyub and McCuen 1997, and Ayyub and White 1978) require ull probabilistic characteristics o both strength and load variables in the limit state equation. or example, the relevant strength variables or stiened panel element are the material s yield strength (stress) y, length o a panel a, thickness t o plating, and dimensions o stiener. While the relevant loads variables are the external pressures due to stillwater bending moment, wave bending moment, and dynamic loads. The deinition o these random variables requires the investigation o their uncertainties and variability. In reliability assessment o any structural system, these uncertainties must be quantiied. urthermore, partial saety actors (PS s) evaluation or both the strengths and loads in any design equation also requires the characterization o these variables. or example, the irst-order reliability method (ORM) as outlined earlier requires the quantiication o mean values, standard deviations (or the coeicient o variation (COV)), and distribution types o all relevant random variables. They are needed to compute the saety index β or the PS s. Thereore, complete inormation on the probability distributions o the basic random variables under consideration must be developed. Quantiication o random variables o loads and strength in terms o their means, standard deviations or COV s, and probability distributions can be achieved in two steps: (a) data collection and (b) data analysis. The irst step is the task o collecting as many sets o data deemed to be appropriate or representing the random variables 15

16 under study. The second is concerned with statistically analyzing the collected data to determine the probabilistic characteristics o these variables. The objective herein is to compile statistical inormation and data based on literature review on both strength and loads random variables or quantiying the probabilistic characteristics o these variables. The quantiication o the probabilistic characteristics o these variables is needed or reliability analysis and design o hull structural components. Tables 11, 12, 13, and 14 provide all the recommended values o inormation required to establish a reliability-based design code or ship structures. This inormation includes limit state unctions or dierent load combinations; probabilistic characteristics (mean values, COV, and distribution type) o random variables involved in these limit state unctions; mean to nominal ratios o these random variables; deterministic values o the probabilistic load-combination actors; mean ratios between dierent load components, ranges o target reliability index; the biases between dierent values o each o the random variables; and probabilistic characteristics o modeling uncertainty. 5.3 Calculations o Partial Saety actors In this section, calculations o partial saety actors (PS s) or both the strength and load components in the limit state unctions or stiened panels are presented herein or demonstration purposes. The irst-order reliability method (ORM) as outlined in Ayyub et al. (2002a) was used to develop the partial saety actors. The partial saety actors are deined as the ratio o the value o a variable in a limit state at its most probable ailure point to the nominal value. The subsequent sections summarize the methods or calculating partial saety actors. They also give a brie review o recommended load and load combinations and their probabilistic characteristics used in computing the partial saety actors. The inal section presents the development o reliability checking or gross panels (grillages) based on stiness o the transverse and longitudinal stieners Perormance unctions or Calculating Partial Saety actors or Stiened Panels Reliability-based design LRD ormat involves the ultimate strength capacity o a stiened plate element and the load random variable o stillwater, wave-induced, and dynamic 16

17 bending moments. The partial saety actors ormat allows transorming the desired reliability index into separate saety actors or each o the design variables in the recommended ormat. Two recommended limit state ormats or stiened panels are provided as ollows: Limit State I: g ( u, WD ) = u kwd WD, (22) Limit State II: (,, ) = k ( k ) g, + (23) u W D u where g = the limit state or perormance unction, = stress due to stillwater bending moment, WD = stress due to combined wave-induced and dynamic bending moments, W = stress due to waves bending moment, k WD = combined wave-induced and dynamic bending moments actor (equals unity), k W = load combination actor equals unity, k D = load combination actor (equals 0.7), and u = ultimate strength capacity o an stiened plate. The ultimate strength capacity u depends on the loading conditions or the stiened panel and is given by the design strength models as described earlier. The two limit states given by Eqs. 22 and 23 are reerred to as limit states 1 and 2, respectively Partial Saety actors or Uniaxial Compression without Lateral Pressure The calculations o the partial saety actors or both limit states given in Eqs 22 and 23 are perormed to provide values or the PS s or all cases such as dierent target reliability levels (3.5, 4.0, and 4.5) and sagging and hogging conditions. These values are rounded to some level deemed to be practical or engineering use. or each case, the values o the PS s beore rounding are denoted by the subscript ( 1 ) as ollows or an example case: and u1 1 1 φ u1 γ + kw W + W W ( γ k γ ) = (24) W1 1 d D1 D D D 1 φ u1 γ + u1 = k γ (25) 1 1 WD WD1 WD 1 17

18 where the above partial saety actors are used as multipliers to the corresponding mean values o the random variables. The ultimate strength capacity o a stiened panel in this case is based on the strength calculated using the Herzog (1987) empirical ormula as discussed earlier. Table 15 provides the recommended load actors applied to the corresponding mean load values based on previously developed LRD guidelines or hull girder bending (Atua 1998) and unstiened panels (Assakka 1998). They are denoted by the subscript ( 2 ) ater rounding. These recommended actors are reerred to as the mean values o the load PS s. The recommended mean values o the load PS s are used to compute the recommended values o the strength actors (applied to the corresponding mean strength values) as ollows or an example case: ( γ W + kdγ D ) W1 D1 ( γ + k ) γ + k W 2 φ = φ u 2 u1 + k γ (26) γ 2 W W W d D D 2 2 and γ + kwdγ WD 2 WD1 φ = φ (27) u2 u1 γ + kwdγ 2 WD WD 2 Table 16 provides the recommended mean values o stiened panel strength actors that are denoted by the subscript ( 2 ). Table 17 provides a summary o the bias actors (mean to nominal ratios) o all random variables involved in the limit states. Based on these bias actors, nominal PS s that can be applied to the corresponding nominal values o the variables, are given by φ = φ B (28) u n u2 ij where B ij = bias actor (mean to nominal ratio). Table 18 provides the recommended nominal values o the load actors, and Table 19 provides nominal values o strength actors. It is to be noted that the values shown in Tables 18 and 19 are rounded which causes a slight change in the implied reliability index, β, according to the LRD guidelines. Thereore, the reliability level calculated or dierent ratios o load components will be slightly greater than the target reliability level or each case which means that the rounded values o the PS s produce slightly saer designs or stiened panels bending and meet the target reliability level. 18

19 5.3.3 Partial Saety actors or Uniaxial Compression with Lateral Pressure The ultimate strength capacity o a stiened panel in this case is based on the strength calculated using the strength model proposed by Adamchak (1997) in Section The procedure or computing the partial saety actors is the same as that used with Herzog s (1987) model except that the value o mean and COV o the bias o the ultimate strength should be revised to account or the variability due to lateral pressure. These values are or the mean and 0.23 or the COV. The general orm o the limit state in this case will be the same as that in Eqs. 24 and 25 except that the lateral pressure eect will be included in the value o the strength reduction actor, φ R. This means that the lateral pressure existence is represented by both the higher value o the COV o the ultimate strength o the stiened panel and the resultant smaller value o φ R (strength value is urther reduced to count or the lateral load). The partial saety actors calculations in this case will be based on the recommended mean load actors (Table 15) and the mean and COV values o the ultimate strength based on Adamchak (1997) model analysis. The resulting recommended mean strength actors in this case are provided in Table 20. The mean /nominal ratio o the strength model used (Adamchak 1997) based on the test results was ound to be The recommended nominal load actors will be the same as those given in Table 18. The resulting recommended nominal strength reduction actors are provided in Table Reliability (Saety) Checking or a Grillage As indicated earlier, the problem o the overall grillage will be reduced to the ailure o the longitudinally stiened sub-panels by preventing the grillage rom buckling as a whole. This is achieved by insuring that the transverse stieners do not delect beyond a certain limit that, in turn, will cause the longitudinal stieners to buckle between the transverse stieners. To perorm reliability checking on the design, the ratio o the stiness o the transverse and longitudinal stieners should not be less than the load eect given by the geometrical parameters shown in the right hand term o the ollowing ormula (Hughes 1988): 19

20 I I y x ( n + 1) 2 2 nπ N 5 3 b a 5 (29) The saety or reliability checking limit state will be reduced to the orm: g I 5 b 1 y ( x) = C 0. 0 I x a (30) where C 1 = panel stiness parameter which depend on the number o bays and stieners. The irst term represents the stiness ratio, and the second term represents the load eect. Our goal here is to develop partial saety actors so that the value o Eq. 30 is not less than zero. The designer in this case will look or the value o the partial saety actors according to his design case (number o bays, number o longitudinal stieners, and the b/a ratio). The minimum required value o the moment o inertia o the transverse stiener should satisy Eq. 30. The partial saety calculations were perormed or dierent design parameters (number o bays, number o longitudinal stieners, and the b/a ratio). However, by examining Eq. 30, it is clear that changing any o these parameters will result in only changing the mean value o the b load eect, C 1. This means that the distribution type and the COV o both stiness ratio a and the load eect will remain the same, i.e., or the same target reliability index, β 0, the same partial saety actors will result or any design case. The dierence will happen only when the COV and distribution type o the stiness ratio change, when the COV and distribution type o the load eect change, or when the target reliability index changes. Dierent design cases were tested to demonstrate the eect o COV and the target reliability index on the PS s. The results are provided in Table 22, which represents the computed partial saety actor that should be multiplied by the stiness ratio to assure the saety criteria o the design concept proposed earlier. However, regardless o the COV s o the b/a ratio or the stiness ratio, the recommended partial saety actors or target reliability levels o 2.5, 3.0, and 3.5, are 0.82, 0.78, and 0.75, respectively, as shown in Table 6. 20

21 5.4 Sample LRD Guidelines This section provides sample reliability-based LRD guidelines or stiened panels and grillages o ship structures. The guidelines, as demonstrated herein, consist o limit state expressions, partial saety actors or both the strength and the loads, and a range o target reliability levels. Stiened plate element o ship structure or all stations should meet one o the limit states as given by Eqs 22 and 23. The ultimate strength capacity u depends on the loading conditions or the stiened panel (i.e., uniaxial, edge shear, etc.) and the strength model that is used. The two limit states given by Eqs. 22 and 23 are reerred to as limit state 1 and 2, respectively. The nominal (i.e., design) values o the strength and load components should satisy these limit states in order to achieve speciied target reliability levels. The strength actors are provided in Table 14 in accordance with the ollowing parameters: (1) target reliability level ranging rom 3.0 to 4.0, (2) the type o load combinations as shown in the table, and (3) ultimate strength prediction or stiened panel as provided by Herzog (1987). The target reliability should be selected based on the ship type and usage. Then, the corresponding strength actor can be looked up rom Table 19 based on the strength model under consideration. The load actors that can be used in conjunction with strength actors are provided in Table 18. or reliability checking on a grillage, Eq. 31 should be used in conjunction with Table 6 to insure that ratio o the stiness o the transverse and longitudinal stieners is met according to ( n + 1) 5 5 I y b φ g (31) I x a nπ + 3 N where I x I y a b n N φ g = moment o inertia o longitudinal plate-stiener = moment o inertia o transverse plate-stiener = length or span o the panel between transverse webs = distance between longitudinal stieners = number o longitudinal stieners = number o longitudinal sub-panels in overall (or gross) panel or grillage = grillages strength reduction actor 21

22 In using the above equation or saety checking or a grillage, a target reliability level should be selected based on the ship type and usage. Then, the corresponding saety actor can be looked up rom Table EXAMPLES DESIGN The ollowing two examples demonstrate the use o LRD-based partial saety in the limit state equation or designing and checking the adequacy o stiened panels o a ship: EXAMPLE 1. Stiened Panel Design Given: A stiened panel, pinned at the ends, whose dimensions are shown in igure 10 is to be designed at the bottom deck o a ship to withstand a uniaxial compression stress due to environmental bending moment loads acting on the ship. The stresses due to the environmental loads are estimated to have the ollowing values: 0.15 ksi due to stillwater bending, 4.5 ksi due to waves bending, and 2.2 ksi due to dynamic bending. I the yield strength o steel is 34 ksi or the plating and 36 ksi or the stiener (i.e., web & lange), and the dimensions o the panel are as shown in Table 23, design the thickness t and length a o the plating assuming a target reliability level o 4.0. Note that the length o the plating is not to exceed 80 in, and not to be less than 48 in. a w t d w t w t b igure 10. Stiened Panel Design 22

23 Table 23. Given Dimensions o the Stiened Panel Variable Value (in) Width o plating, b 24.0 Stiener web depth, d w 4.50 Stiener lange breadth, w 1.75 Stiener web thickness, t w Stiener lange thickness, t Solution: or stiened panel under uniaxial compression without lateral pressure, the strength model as given by Eq. 7 (Herzog) applies u m = m y y ka rπ ka rπ E y E y b t Assume an initial value or t = 0.2 in, and or a = 80 in, hence y A A = p s = bt = 24(0.2) = 4.8 in = t ys A A s s w + t d + + A Check the slenderness ratio b/t: b t = yp p w A p w 2 = 0.375(1.75) (4.5) = in 36(1.579) + 34(4.8) = = ksi = 120 > 45, thereore, the ollowing equation applies: u ka = y 1 b m y rπ E t 2 or or b 45 t b t > 45 The radius o gyration r or the cross section can be ound when the moment o inertia I has been established. To compute I, the location o neutral axis y must be calculated: y = = the plating. ( 24)( 0.2) ( 4.5)( 0.205) ( 0.375)( 1.75) in. rom the base o

24 Thereore, I = in 4 I 17.23, and r = = = in A Assuming m and k both equal to one (see Tables 2 and 3), we have u = ( 1)(34.50) = ksi (1.65) 29, π In reerence to Tables 18 and 19, and or a target reliability index β 0 = 4.0 as given, the ollowing partial saety actors are obtained or use in the design equation: φ = 0.57, γ = 1.05, γ W = 1.7, and γ D = 1.1 Thereore, φ u = 0.57(12.03) = 6.86 ksi γ + k W (γ W W + γ D k D D ) = (1.05) (0.15) + (1) [1.7(4.5) + (1.1) (0.7) (2.2)] = 9.50 ksi (φ u = 6.86 ksi ) < 9.84 ksi Not Acceptable Now try t = 0.25 in and a =80 in, hence, y A A = p s = bt = 24(0.25) = 6 in = t ys A A s s w + t d + + A Check the slenderness ratio b/t: b t = yp p w A p w 2 = 0.375(1.75) (4.5) = in 36(1.579) + 34(6) = = ksi = 96 > 45, thereore, the ollowing equation applies: u ka = y 1 b m y rπ E t Again, the radius o gyration r or the cross section can be ound when the moment o inertia I is established. To compute I, the location o neutral axis y must be calculated: y = = the plating ( 24)( 0.25) ( 4.5)( 0.205) ( 0.375)( 1.75) 0.831in. rom the base o

25 Thereore, I = in 4 I 18.22, and r = = = in A Assuming m and k both equal to one (see Tables 2 and 3), we have u = ( 1)(34.42) = ksi (1.55) 29, π φ u = 0.57(15.87) = 9.05 ksi γ + k W (γ W W + γ D k D D ) = (1.05) (0.15) + (1) [1.7(4.5) + (1.1) (0.7) (2.2)] = 9.50 ksi (φ u = 9.05 ksi ) < 9.50 ksi Not Acceptable Now try t = 25 in and a = 60 in. Thereore, in this case, the section properties calculations (i.e., y, I, and r) will be the same. However, the strength will change due to a new value o a = 60 in: u = ( 1)(34.42) = ksi (1.55) 29, π φ u = 0.57(17.43) = 9.94 ksi γ + k W (γ W W + γ D k D D ) = (1.05) (0.15) + (1) [1.7(4.5) + (1.1) (0.7) (2.2)] = 9.50 ksi (φ u = 9.94 ksi ) > 9.50 ksi Acceptable Hence, select t = 0.25 in, and a = 60 in EXAMPLE 2. Adequacy Checking or Grillage Given: Assume a target reliability level o 2.5, check the adequacy o the ollowing grillage: I x = 16 in 4, I y = 26.5 in 4, N = 5, n = 3, a = 60 in, b =24 in Solution: or a grillage, the strength is given by Eq. 31 as φ g I I y x ( n + 1) 2 2 nπ N 5 3 b a or target reliability index o 2.5, Table 6 gives φ g = 0.78, thereore, 5 25

26 φ g I I y x 26.5 = 0.78 = ( n + 1) b ( 3 + 1) 2 2 nπ N 3 a = 2 2 (3π ) (5) Since 1.29 < 1.33, the grillage will be inadequate. 5 = SUMMARY AND CONCLUSIONS uture design guidelines or stiened panels and grillages o ship structures will be developed using reliability methods and they will be expressed in a special and practical ormat such as the Load and Resistance actor Design (LRD). The LRD guidelines or stiened panels, which are based on structural reliability theory, can be built on previous and currently used speciications or ships, buildings, bridges, and oshore structures. This paper provides methods or and demonstrates the development o LRD guidelines or ship stiened panels and grillages elements subjected to uniaxial loading. These design methods were developed according to the ollowing requirements: (1) spectral analysis o wave loads, (2) building on conventional codes, (3) nominal strength and load values, and (4) achieving target reliability levels. The irst-order Reliability Method (ORM) was used to develop the LRD-based partial saety actors (PS s) or selected limit states and or various types o loading acting on unstiened panel element. These actors were determined to account or the uncertainties in strength and load eects. ORM was used to determine these actors based on prescribed probabilistic characteristic o strength and load eects. Also, strength actors were computed or a set o load actors to meet selected target reliability levels or demonstration purposes. The resulting LRD guidelines are demonstrated in this paper using examples design. ACKNOWLEDGMENTS The authors would like to acknowledge the opportunity and support provided by the Carderock Division o the Naval Surace Warare Center o the U.S. Navy through its engineers and researchers that include J. Adamchak, J. Beach, T. Brady, D. Bruchman, J. Dalzell, A. 26

27 Disenbacher, A. Engle, B. Hay, D. Kihl, R. Lewis, W. Melton, W. Richardson, and J. Sikora; and the guidance o the Naval Sea System and Command by E. Comstock, J. Hough, R. McCarthy, N. Nappi, T. Packard, J. Snyder, and R. Walz. REERENCES 1. Ang, A. H.-S., and Tang, W. H., Probability Concepts in Engineering Planning and Design, Vol. II Decision, Risk, and Reliability, John Wiley & Sons, NY. 2. Assaka, I., Ayyub, B. M., Hess, P., and Atua, K., 2002, Reliability-Based Load and Resistance actor Design (LRD) Guidelines or Unstiened Panels, Naval Engineers Journal, ASNE, 114(2). 3. Assakka, I. A and Ayyub, B. M., An Excel Spread Sheet or Reliability Assessment and Partial Saety actors Calculations, the Center or Technology and Systems Management, Department o Civil and Environmental Engineering, University o Maryland, College Park, MD. 4. Assakka, I. A. and Atua, K. I., A Spreadsheet Calculations or Evaluating the Strength o Stiened Panels o Naval Ships. the Center or Technology and Systems Management, Department o Civil and Environmental Engineering, University o Maryland, College Park, MD. 5. Assakka, I. A., "Reliability-based Design o Panels and atigue Details o Ship Structures," A dissertation submitted to the aculty o the Graduate School o the University o Maryland, College Park in partial ulillment o the requirements or the degree o Doctor o Philosophy. 6. Atua, K. I., "Reliability-Based Structural Design o Ship Hull Girders and Stiened Panels," A dissertation submitted to the aculty o the Graduate School o the University o Maryland, College Park in partial ulillment o the requirements or the degree o Doctor o Philosophy. 7. Ayyub, B. M. and McCuen, R. H., Probability, Statistics And Reliability or Engineers, CRC Press, L. 8. Ayyub, B. M. and White, A. M., Reliability-Conditioned Partial Saety actors, Journal o Structural Engineering, Vol. 113, No. 2, ebruary, ASCE,

28 9. Ayyub, B. M., Assakka, I., Beach, J. E., Melton, W., and Conley, J. A., 2002a, Methodology or Developing Reliability-Based Load and Resistance actor Design (LRD) Guidelines or Ship Structures, Naval Engineers Journal, ASNE, 114(2). 10. Ayyub, B. M., Assakka, I., Sikora, J., Adamchack, J., and Atua, K., 2002b, Reliability- Based Load and Resistance actor Design (LRD) Guidelines or Hull Girder Bending, Naval Engineers Journal, ASNE, 1142(2). 11. Bleich,., Buckling Strength o Metal Structures, McGraw-Hill, NY. 12. Bruchman, D. and Dinsenbacher, A., Permanent Set o Laterally Loaded Plating: New and Previous Methods, SSPD , David Taylor Research Center, Bethesda, Maryland. 13. aulkner, D., A Review o Eective Plating or Use in the Analysis o Stiened Plating in Bending and Compression, Journal o Ship research, 19(1), aulkner, D., Compression Strength o Welded Grillages, Chapter 21 in Ship Structural Design Concepts, editor J. N. Evans, Cornell Marine Press. 15. rieze, P. A., Dowling, P. J., and Hobbs, R. W., Ultimate Load Behavior o Plates in Compression, International Symposium on Steel Plated Structures, Crosby Lockwood Staples, London. 16. Hughes, O.., Ship Structural Design, A rationally-based, Computer-Aided Optimization Approach, The Society o Naval Architects and Marine Engineers, Jersey City, New Jersey. 17. Mansour, A.E., Jan, H. Y., Zigeman, C., I., Chen, Y. N., Harding, S. J., Implementation o Reliability Methods to Marine Structures, Report, The Society o Naval Architects and Marine Engineering, Mansour, A.E., Wirsching, P.H., and White, G.J., and Ayyub, B. M., Probability- Based Ship Design: Implementation o Design Guidelines, SSC 392, NTIS, Washington, D.C., 200 pages. 19. Sikora, J. P., Dinsenbacher, A., and Beach, J. A., A Method or Estimating Lietime Loads and atigue Lives or Swath and Conventional Monohull Ships, Naval Engineers Journal, ASNE,

29 20. Soares, C. G., Uncertainty Modeling in Plate Buckling, Shipbuilding Engineering Program, Department o Mechanical Engineering, Technical University o Lisbon, Elsevier Science Publishers B. V., Amsterdam, Printed in The Netherlands. 21. Valsgard, S., Numerical Design Prediction o the Capacity o Plates in Biaxial In- Plane Compression, Computers and Structures, 12,

30 Gross Panel Keel Longitudinally Stiened Sub-Panel igure 1. Portion o the Hull Girder Showing the Gross Panel (i.e., Grillage) and a Longitudinally Stiened Sub-Panel (Hughes, 1988) Correlation actor Ship's Length in t Sagging Hogging igure 2. Correlation Coeicient o Whipping Bending Moment (k D ) or 300 < LBP < 1000 t (Mansour et al and Atua. 1998) 30

31 igure 3. Interaction Diagram or Collapse Mechanism o a Stiened Panel under Lateral and Inplane Loads (Hughes 1988) 31

32 igure 4. Instability ailure Modes (Adamchak 1979) 32

33 igure 5. Types o Beam-column ailure (Adamchak 1979) igure 6. Variation o Bias o Strength Models as a unction o Slenderness Ratio o Column Under Uniaxial Load Only (Assakka 1998 and Atua 1998) 33